The shrinking figure eight and other solitons for the curve diffusion flow

TL;DR

This paper studies special solutions to the curve diffusion flow, classifying stationary solutions, analyzing shrinkers, translators, rotators, and explicitly parametrizing a shrinking figure eight curve.

Contribution

It provides a complete classification of stationary solutions and explicit parametrization of a shrinking figure eight for the curve diffusion flow.

Findings

Complete classification of stationary solutions

Qualitative analysis of shrinkers, translators, rotators

Explicit parametrization of a shrinking figure eight

Abstract

In this article we investigate the dynamics of special solutions to the surface diffusion flow of idealised ribbons. This equation reduces to studying the curve diffusion flow for the profile curve of the ribbon. We provide: (1) a complete classification of stationary solutions; (2) qualitative results on shrinkers, translators, and rotators; and (3) an explicit parametrisation of a shrinking figure eight curve.